I found the picture but I painted very wrong .
I have no idea what 'linear' means in the above picture. You would need to provide information from wherever the picture came from about the definition.
As for GR, I disagree in that velocity of distant objects if fundamentally ambiguous in GR. First, you can talk about two different kinds of thing:
1) growth of proper distance between objects as function of time. This depends on how you foliate spacetime, and how you assign time to each slice. This is summarized as 'coordinate choice dependent'. There is a coordinate choice that is strongly preferred [for good reasons] by cosmologists, but note cosmology is just a special case of general relativity, and you can't even define such coordinates for an arbitrary solution. Even with preference, that still doesn't alter the fact that this quantity is influenced by a choice not required by the theory. And any other choice of coordinates will lead to the same predictions. This type of velocity can exceed c, even in special relativity.
2) Relative velocity, obtained by comparing velocity vectors. In SR, this is unambiguous, and relative speed is invariant. However, in GR, the result of comparing vectors depends on how you bring them together, and you get different answers for different paths by which you try to move them together. However, in GR, no matter what path you use, the relative speed will always be less than c.
Thus, you have a choice between a coordinate dependent quantity and a path dependent quantity, thus fundamentally ambiguous.
What is not ambiguous is prediction of red shift amount. This is what is best to focus on, rather than pretending there is anything resembling a well defined velocity between distant galaxies.
But you still didn't explain where it came from or what it means or give a link to a scientific source. I'm closing this thread; if you can give an actual reference, PM me and I'll reopen it.
[Edit: Thread reopened.]
Link to picture:
I was thought that in general relativity speed can exceed c (according to graph ) but it was redshift- velocity diagram.In general relativity speed cant exceed c isnt it ?
The diagram is actually misleading, because "velocity" means three different things for the three different curves.
(1) For the "linear" curve, "velocity" is just the redshift times ##c##. This tells you nothing useful physically; it's just a convention that cosmologists sometimes use to quote redshifts in units of velocity.
(2) For the "special relativity" curve, "velocity" is the coordinate velocity of the object emitting light which is observed to have redshift ##z##, in an inertial frame in which the observer is at rest, assuming spacetime is globally flat.
(3) For the "general relativity" curve, "velocity" is the recession velocity of a "comoving" object emitting light which is observed to have redshift ##z## by a "comoving" observer, relative to the observer. This recession velocity is obtained by multiplying the Hubble constant by the proper distance, so it's not a coordinate velocity. (There are actually a range of "GR" curves, depending on what assumptions you make about the dynamics of the universe's expansion.)
Of the above three things, only (2) can't exceed ##c##. More generally, in curved spacetime, locally measured velocities (i.e., velocities that can be measured entirely within a single local inertial frame) can't exceed ##c##. But pretty much anything else that gets called "velocity" (often misleadingly, as above) can exceed ##c## in GR.
Ok, with context, my prior post explains it all. The linear case is just a strawman theory, since it posits linear relation between redshift and speed. The diagram is otherwise misleading because (as labeled), it is using recession speed for GR and relative speed for SR, which is apples and oranges. Unfortunately, this is a very common mistake in the literature. You can draw an SR curve with recession velocity limiting at 2c just by using a different inertial frame (because recession velocity is coorinate dependent). Further, if you use a cosmological style of coordinates in SR (look up Milne coordinates or model), you can draw a curve for SR that looks like the GR curve. In my view it is simply long standing error in the literature that attaches the wrong significance to cosmological recession velocity.
Please study my prior post.
Also watch this for more explanation of recession velocity vs velocity:
This is exactly the 'gee whiz' interpretation I disagree with. And the fundamental proof is the Milne cosmology, which is pure flat spacetime yet has all these allegedly strange features. (It obviously fails as a predictive model but it shows that most of the features considered unique to GR cosmology are present in SR, and are really features of coordinate choice, or to more physically justify it, the congruence representing the main matter flow). When the video uses terms lime 'moving away from us faster than light', it is using relative velocity terminology for what should be considered a recession velocity: how distance is growing between two objects in mutual relative motion described from a common coordinate system in which their motion is isotropic - which leads to 2c trivially using even Minkowski coordinates.
So my sum, is bad video, reinforcing several prevalent but incorrect concepts.
Relative velocity is ambiguous (comparison path dependent) but always < c in GR, no exceptions mathematically possible. Recession velocity can be arbitrarily greater than c, in both SR and GR, and is coordinate dependent. In cosmology, the coordinate choice used is physically well motivated by the matter flow: coordinates that make the isotropy and homogeneity explicit. But this doesn't change the mundane interpretation of recession velocity as something that can exceed c locally in SR.
red shift can be caused by any of the following three things:
(1) relative velocity between objects; (2) a change in the size of space which changes the size of objects in space, and (3) by chages in the rate of time of the observer since the photon was emitted.
These are impossible to separate or even define in a general spacetime (only stationary spacetimes have a potential you can use to define a 'rate of time' as a function of position; only isotropic, homogeneous cosmological solutions have an expansion of space, and it is only relative to a particular choice of how to foliate spacetime). Meanwhile, you can take any of 3 pure SR formulations of red shift, generalize them to curved spacetime, and they apply to all GR solutions without exception. There is never, ever, a need to consider different types or sources of redshift except as a way to simplify analysis.
The 3 universal red shift formulations that apply to all cases in both SR and GR, each of which completely specifies redhsift are:
1) Connect emission event on emitter world line to reception event on receiver world line via a null geodesic. Do so again for an infinitesimal moment later on the emitter world line. The ratio of proper times between these events on the receiver wold line and the emitter world line is the Doppler factor.
2) Using the emitter 4-velocity at event of emission and the receiver 4-velocity at event of reception, apply the SR relative velocity Dopper formula. In the case of curved spacetime, you have to parallel transport the emitter 4-velocity along the same connecting null geodesic as in (1) [in SR, parallel transport is path independent so you don't need to specify any trasnsport path]
3) The above are pure geometry. There is universal kinematic approach as well. Compute the 4-momentum of the light as emitted. Parallel transport along the null geodesic as in (1) to the reception event [in SR, a path need not be specified]. Take the dot product of the transported 4-momentum with receiver 4-velocity at reception event. This, compared to the same dot product at emission event gives you the Doppler factor.
Note, there is no mention of potential, 'time rate as function of position', or expansion of space, in any of these formulations, yet each is universally sufficient (by itself) in any GR solution. [In case it isn't clear, I should say that all three always give the same answer].
[Mathematically, all of these procedures directly use only the following geometric objects in the region of interest:
a) The metric at emission and reception events (for either computation of proper time or dot products).
b) The connection coefficients between these events (for specification of the null geodesic, and for parallel transport) ]
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